Question

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

$\dfrac{1}{2187}$

$\dfrac{1}{729}$

$\dfrac{2}{243}$

$\dfrac{1}{81}$

$\dfrac{5}{243}$

Solution:

After $7$ moves there are $3^7 = 2187$ equally likely walks. A successful walk visits every vertex exactly once.

From the start there are $3$ choices for the first move and $2$ for the second (not returning). Labeling the first three vertices $A, B, C,$ the bug must move to one of two vertices, after which the route is forced except for a single binary choice, giving $3 \cdot 2 \cdot 3 = 18$ such paths.

The probability is $\frac{18}{2187} = \frac{2}{243}.$

Thus, the correct answer is C.

2006 AMC 10A Problem 25

Solution:

Problem 25 in Other Years